Question: Solve for $x$ and $y$ using elimination. $\begin{align*}-5x+y &= -1 \\ 9x-9y &= 5\end{align*}$
Explanation: We can eliminate $y$ when its corresponding coefficients are negative inverses. Recalling our knowledge of least common multiples, multiply the top equation by $9$ and the bottom equation by $1$ $\begin{align*}-45x+9y &= -9\\ 9x-9y &= 5\end{align*}$ Add the top and bottom equations. $-36x = -4$ Divide both sides by $-36$ and reduce as necessary. $x = \dfrac{1}{9}$ Substitute $\dfrac{1}{9}$ for $x$ in the top equation. $-5( \dfrac{1}{9})+y = -1$ $-\dfrac{5}{9}+y = -1$ $y = -\dfrac{4}{9}$ $y = -\dfrac{4}{9}$ The solution is $\enspace x = \dfrac{1}{9}, \enspace y = -\dfrac{4}{9}$.